Showing posts with label
School Mathematics from Higher Viewpoints.
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Showing posts with label
School Mathematics from Higher Viewpoints.
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<The previous article in this series | The table of contents of this series |
A same length is measured 1.00 m, 3.28 ft, e.t.c., so what?
Topics
About:
elementary school mathematics
The table of contents of this article
Starting Context
Target Context
-
The reader will know the relation between the reality and measurements.
Orientation
There is an article on becoming a benefactor of humanity by being a conduit of truths
Main Body
1: 'Measurement' Is a Basis for Applying Mathematics to the Reality
Special-Student-7-Hypothesizer
As I look at a Japanese elementary school mathematics curriculum, children 1st learn natural numbers with their additions, subtractions, and multiplications, and then, their divisions together with fractions and decimal fractions, but intermingled with the process, children learn measurements.
Special-Student-7-Rebutter
Theoretically speaking, learning 1st collections (and maybe sets), the natural numbers set, the integral numbers set, the rational numbers set, and the real numbers set seems logical, but the curriculum does not go that way.
Special-Student-7-Hypothesizer
In fact, children do not learn negative numbers until junior high school, and as far as I judge from the curriculums for elementary school, junior high school, and high school that I am looking at right now, children do not seem to learn collections until high school.
Special-Student-7-Rebutter
Really? How could children learn functions or probabilities without the concept of collection?
Special-Student-7-Hypothesizer
I do not know; I am looking only at the curriculums not at any textbook, so, the concept of collection may be being cursorily introduced somewhere mingled in a topic.
Anyway, I wanted to say that 'measurement' is being introduced at the 1st stage of mathematics educations.
Special-Student-7-Rebutter
Well, 'measurement' does not seem any indispensable part of mathematics, or is 'measurement' really a part of mathematics?
Special-Student-7-Hypothesizer
We are not talking about the measure theory in mathematics, which is about measuring some subsets of a set, while 'measurement' here is about measuring the reality, which means appointing a number to the reality.
Special-Student-7-Rebutter
'measurement' here does not seem any part of pure mathematics, but is a basis of applying mathematics to the reality.
Special-Student-7-Hypothesizer
The reason why 'measurement' is learned at the 1st stage is that the school mathematics is not meant to teach pure mathematics but to teach applied mathematics.
Children learn natural numbers in order to count some apples (counting is a kind of measurement) and learn decimal fractions in order to measure the length of a side of a desk.
So, school mathematics does not follow the logical order of the construction of pure mathematics but jump to some urgent applications and conveniently pick up some mathematical entities necessary for the applications.
2: Measurement Is to Represent an Aspect of the Reality by a Mathematical Entity
Special-Student-7-Hypothesizer
In general, 'measurement' is to represent an aspect of the reality by a mathematical entity (typically, a number).
For example, for the reality in which a box contains some apples, a measurement gives 5.
I said "an aspect", because there are many boxes and many non-boxes in the reality and the measurement looked at only the specific box and cared only about the quantity of what are inside the box: we cannot exactly represent the whole reality by anything.
So, not only measuring some lengths by some rulers and measuring some weights by some scales but also countings are measurements.
3: Let Us Measure a Length
Special-Student-7-Hypothesizer
Let us measure the length of a side of a desk.
Special-Student-7-Rebutter
Do we need to?
Special-Student-7-Hypothesizer
Ah, that may sound like just a guff, but is really an important question; certainly, we do not need to measure the length, at least, as far as the desk is concerned: the desk exists without being measured, or more generally, the reality exists without being measured. We will measure the length just because that suits our convenience.
Special-Student-7-Rebutter
I am making sure of that point, because there are some people who are claiming "There is no reality except measurements.".
Special-Student-7-Hypothesizer
According to that claim, the desk did not exist before the human measured the desk, and at the instant the human measured the desk, the desk popped up in the reality, and furthermore, at the instant the human abandoned measuring the desk, the desk puffed out from the reality, ..., is that any serious talk?
Special-Student-7-Rebutter
Those people are adamant.
Special-Student-7-Hypothesizer
... Let us talk about it later.
Anyway, we use a ruler to measure the length.
Special-Student-7-Rebutter
Or a tape measure.
Special-Student-7-Hypothesizer
Yes, but that is not any issue here, so, let us call it a ruler.
An issue is that the ruler may be a meter-ruler, an yard-ruler, or any some-other-unit-ruler.
Accordingly, the measurement may be 0.91 (m), 1.00 (yard), or something.
Special-Student-7-Rebutter
With an error.
Special-Student-7-Hypothesizer
Yes, with an error, because the ruler is not so accurate (the production is not so perfect) and is deformed somewhat by temperature, moisture, gravity, e.t.c., your eyesight is not so accurate, e.t.c..
Students learn that it is crucial to specify the possible error in any measurement.
Anyway, the measurement depends on the unit you choose.
So, measurements are relative with respect to units.
But that does not mean that the reality is relative.
In fact, if you have changed from meter to yard, the measurement has changed, but the length is not changed just because you have chosen a fancy unit.
Special-Student-7-Rebutter
The seed of misleading is the definition of 'length'.
Special-Student-7-Hypothesizer
To simplify the argument, let us forget Relativity throughout this section.
Then, I mean by 'length' how much the 2 points are separated in the space in the reality.
Just because you have used a yard-ruler, the 2 points have not become less separated.
In fact, if you thought that the desk had shrunk just because you used a yard-ruler, you would be crazy; or when a man lives 1,000.000 km separated from his love, if he thought that adopting a unit, perhaps called "yearn", by which he was 0.000,001 yearn separated from his love, would shrink the distance to his love from him, he would be a romanticist.
Likewise, measurements' having some errors does not mean that the reality has some errors.
If you thought that the reality had an error just because the production of the ruler was not perfect and your eyesight was not accurate, you would be crazy, and egocentric, in addition.
Special-Student-7-Rebutter
I talked about "The seed of misleading", because if someone meant by "length" 'measurement of the length', "lengths" would be relative with respect to units.
Special-Student-7-Hypothesizer
We mean by "length" 'how much the 2 points are separated in the space in the reality', and when we mean 'measurement of the length', we use "measurement of the length".
In fact, calling 'measurement of the length' "length" is a sloppy abbreviation, and some people refuse to refrain from using such sloppy abbreviations, which causes confusions.
When a measurement of the length of a side of a desk is 1.00 m and a measurement of the length of a side of another desk is 1.00 yard, we do not say "The 2 desks have the same length.".
4: But How Does Relativity Change the Situation?
Special-Student-7-Rebutter
It is good so far for non-Relativistic cases, but how does Relativity change the situation?
Special-Student-7-Hypothesizer
Relativity does not say "the reality is relative" at all, although it is rather prevalently misunderstood to be saying so.
Special-Student-7-Rebutter
What does Relativity say about "the length of the side of the desk"?
Special-Student-7-Hypothesizer
Relativity says that the universe is a 4-dimensional spacetime.
Note that the spacetime itself is not relative in any way.
But when you begin to talk about a "space", what is the "space"?
Special-Student-7-Rebutter
The "space" seems to mean a 3-dimensional space.
Special-Student-7-Hypothesizer
When you talk about a 3-dimensional "space", you mean a cross section of the 4-dimensional spacetime.
But you can choose infinitely many cross sections of the 4-dimensional spacetime.
So, the "space" depends on your choice, which means that "space"s are relative with respect to your choices.
If someone finds difficult to imagine the 4-dimensional spacetime, he or she could imagine the spacetime as a 2-dimensional plane, as a metaphor.
Then, a "space" would be a line of your choice on the plane.
The 2 ends of the side of the desk draw 2 so-called "world line"s in the spacetime, and when you choose a "space", each of the world lines intersects the "space" at a point, and when you say "how much the 2 points are separated in the space in the reality", "the 2 points" are the 2 intersections.
As the 2 intersections depend on your choice of the "space", the length depends on your choice of the "space", which is what Relativity says.
Special-Student-7-Rebutter
Whatever "space" you choose, that does not change the reality at all, which is parallel to that choosing a unit of length does not change the reality.
Special-Student-7-Hypothesizer
'choosing a "space"' is typically related with choosing a 3-dimensional coordinates system: the 3-dimensional coordinates system at a "time" is really a cross-section of the 4-dimensional spacetime, which is called "space", and the 4-dimensional spacetime is sliced into such parallel "space"s with different "time"s.
And some 2 3-dimensional coordinates systems moving against each other have sliced the 4-dimensional spacetime in some different directions.
Special-Student-7-Rebutter
Is it not bad to adopt a 3-dimensional coordinate system for the 4-dimensional reality?
Special-Student-7-Hypothesizer
In fact, it is misleading if not absolutely bad: people tend to imagine as though "the space" is developing through "time", but your "space" is just your choice, not any objective existence.
Now, you should understand why the 2 coordinates systems that are moving against each other measure the side of the desk differently.
In the 2-dimensional spacetime metaphor, the 2 ends of the side of the desk draw some parallel world lines on the 2-dimensional plane (let the plane be the x-y plane and let the 2 world lines be x = 1 and x = 2), and the 1st coordinates system is the bunch of y = t lines and the 2nd coordinates system is the bunch of y = x + t' lines.
Then, the 1st coordinates system sees at t = 0, the 2 ends of the side of the desk at (1, 0) and (2, 0), while the 2nd coordinates system sees at t' = 0, the 2 ends of the side of the desk at (1, 1) and (2, 2).
So, the 1st coordinates system talks about the length between (1, 0) and (2, 0) and the 2nd coordinates system talks about the length between (1, 1) and (2, 2), and it is not mysterious that they report some different measurements, because the 2 coordinates systems are talking about the different pairs of points.
Special-Student-7-Rebutter
As a caveat, the 4-dimensional spacetime has the Lorentzian metric, not the Euclidean metric.
Special-Student-7-Hypothesizer
Yes, "the Euclidean metric" means that the length between \((x_1, x_2, x_3, x_4)\) and \((x'_1, x'_2, x'_3, x'_4)\) is \(\sqrt{(x'_1 - x_1)^2 + ... + (x'_4 - x_4)^2}\), but "the Lorentzian metric" does not say so (we will not discuss here what exactly it says).
So, in the 2-dimensional metaphor, the length between (1, 0) and (2, 0) is not necessarily 1 and the length between (1, 1) and (2, 2) is not necessarily \(\sqrt{2}\).
So, we are not saying "the length between (1, 1) and (2, 2) is larger than the length between (1, 0) and (2, 0)" but are saying just that it is natural that the 2 coordinates systems report different measures.
Special-Student-7-Rebutter
But the reality as the 2-dimensional plane is not concerned with what coordinates systems a crazy human has chosen or what measurements are with respect to such coordinates systems.
Special-Student-7-Hypothesizer
As some 2 rulers with different units measure the desk differently, some 2 rulers moving against each other measure the desk differently, which is no mystery: it is about that just like the 2 rulers with different units are different rulers, the 2 rulers moving against each other are different rulers, and it is natural that some different rulers measure differently.
In fact, an observer (a coordinates system) flies by the earth with a high speed relative to the earth and measures the length between the 2 lovers as 1 cm, but how is that a consolation for the poor romanticist?
5: The Reality Is No Uncertain
Special-Student-7-Hypothesizer
Another prevalent misunderstanding is "the reality is uncertain, by the Heisenberg uncertainty principle".
No.
The Heisenberg uncertainty principle is claiming that any simultaneous measurement of each of some pairs of observables is uncertain.
Special-Student-7-Rebutter
What are uncertain are some measurements not the reality.
Special-Student-7-Hypothesizer
And there is also a prevalent misunderstanding that the break of Bell's inequality has proved "the uncertainty of the reality".
No.
The break of Bell's inequality is also suggesting the uncertainty of some measurements.
Special-Student-7-Rebutter
We need to distinguish the reality and its measurements.
Special-Student-7-Hypothesizer
The reality exists without being measured and does not need to be measured.
For example, did the Big Bang need to be measured for it to exist?
Was the Big Bang uncertain until a human measured it?
Special-Student-7-Rebutter
Has the Big Bang been measured even now?
Special-Student-7-Hypothesizer
It seems impossible to fully measure the Big Bang, while some humans have measured a tiny part of the aftermath of the Big Bang.
So, does the Big Bang become somewhat a little more certain at the instant a human measures a tiny part of the aftermath, retroactively?
Special-Student-7-Rebutter
In fact, any measurement is really an aftermath of what was intended to be measured in the reality.
Special-Student-7-Hypothesizer
When a human tries to measure an aspect of the reality, the human puts an apparatus near the aspect, and the apparatus and the aspect interact somehow, and the result of the interaction is the measurement.
If you claim that the aspect is determined retroactively because of the interaction, that is a violation of causality.
Special-Student-7-Rebutter
I thought that even quantum mechanics did not refute causality.
Special-Student-7-Hypothesizer
If the laws that govern the reality are probabilistic, that interaction is probabilistic, and the measurement is not the exact representation of the aspect.
Bell's inequality is based on the assumption that measurements are the exact representations of the aspects of the reality, and its break is suggesting that the assumption is wrong.
Note that Bell's inequality was introduced for the purpose of supporting Mr. Einstein's claim, "God does not play at dice", and naturally supposed that the laws of the reality were deterministic, which would imply that measurements as interactions were deterministic, which the results seem to have refuted.
Although there are some prevalent misstatements like "The reality is not determined until it is measured.", absolutely no, 'measurements are not determined until the measurements are performed' seems a more correct statement, which is natural because any measurement is a physical phenomenon and as far as the laws of the reality are probabilistic, the outcome of the measurement is not determined until the outcome happens.
6: Measurements Are Harassments
Special-Student-7-Hypothesizer
Let us think of an allegory.
You, an Earthian, are captured by an alien species.
The alien species feel only 'fear' and 'anger', and so, they understand only 'fear' and 'anger', in fact, they cannot ever imagine whatever else except 'fear' and 'anger' exist in the reality.
You feel 'love', 'pity', 'joy', 'sadness', e.t.c., I hope.
The alien species measure you, which is to determine whether you are in 'fear' or in 'anger', for the alien species, because that is all they can ever imagine to exist.
So, the alien species harass you to force you to show 'fear' or 'anger', and they are satisfied when you show 'fear' or 'anger' saying "Oh, this maggot was in fear!".
They invent Bell's inequality and conclude that your state was not determined until they measured you, and they even begin to claim "We determine the reality!".
Special-Student-7-Rebutter
Of course, you were not in fear until they measured (harassed) you.
Special-Student-7-Hypothesizer
The alien species's conceptions are only "in fear or in anger", and when you are in pity, they declare "This maggot's state is uncertain." and claim "The reality is not determined until we measure it.".
Special-Student-7-Rebutter
"state" by them is the state brought about by their harassment, and certainly, the "state" does not appear in the reality until they perform the harassment.
Special-Student-7-Hypothesizer
And if your reaction to the harassment is probabilistic, the "state" was certainly not determined until they performed the measurement.
But that does not mean you had not been in an absolutely definite state of pity before the harassment was performed.
Special-Student-7-Rebutter
We should be aware that human concepts like "position", "spin", e.t.c. are derived from pathetically limited human daily experiences, and Earthians who are trying to measure the reality in such concepts are like the alien species who are trying to measure you in 'fear' or 'anger'.
7: Isn't There a Misunderstanding About 'Scientific Method'?
Special-Student-7-Hypothesizer
Isn't there a misunderstanding about 'scientific method'?
Certainly, science values and requires measurements, because just willfully claiming something without any measurement does not give any credibility.
But science is also based on the assumption of the existence of the objective reality.
In fact, if there was no objective reality, what would we measure?
I thought that as we supposed the existence of the objective reality, we were trying to measure the reality. If measurements are relative and uncertain, that is not convenient for us, but we must accept it, and we try to approach the objective reality as near as possible, if not we reach the objective reality, through trial-and-errors of making hypotheses and checking them with measurements, which is the scientific method, I thought.
The scientific method is never "There is no reality except measurements.", I thought.
Special-Student-7-Rebutter
If "There is no reality except measurements", why don't we just do willful measurements to get desired results? We would not need to pay any respect to non-existent reality, so, any willful measurement would be as good as any measurement.
Special-Student-7-Hypothesizer
'accuracy' is the difference between the objective reality and the measurement, and if there was no objective reality, we would not need to care about any accuracy, because there would not be such thing as 'accuracy', because there would be not anything with which the measurement is compared for accuracy.
So, we would concoct an apparatus that would give '1' anyway, and we would find a golden theory, "Measurements are always '1'.". ... Why not? As only we were creating the reality, why wouldn't we create the willful reality?
Or you should just stop any measurement, and there would be no reality, and "There is no reality." would be the absolutely complete theory.
Special-Student-7-Rebutter
Sun will burn out in short time, Earthians will perish with it, and just because pathetic Earthians will perish on a pathetic remote tiny planet, will the reality disappear because measurements will not be performed anymore?
8: A Bad Moral Implication
Special-Student-7-Hypothesizer
In fact, "There is no reality except measurements." is the root of every evil.
"More people came to my inauguration than to the predecessor's!" would be an "alternative truth", because that perception was a measurement and there would be no objective reality to refute the measurement.
There would be no lie, because there would be no objective reality to compare with for the accuracy of the statement.
So, a villain would claim anything willfully with indemnity.
Sounds familiar?
Blackguards cite some misleading YouTube videos or TV programs that claim that Relativity and Bell's inequality have proved "the relativity and uncertainty of the reality" to justify themselves, saying "So, there is no objective truth, so, whatever I say are as truthful as whatever anyone says.".
...
Instead, we are supposed to accept the existence of the objective reality and be humble that our present claim is somehow relative and uncertain, so, we need to be ready to listen to claims by others to remedy our claim to make it more impartial.
Special-Student-7-Rebutter
Relativity and Bell's inequality are in fact some opportunities for us to be humble if they are understood properly, but with their understood improperly (via improper promotions), they are enticing absolute arrogance.
9: Conclusion
Special-Student-7-Hypothesizer
After all, what good children need to engrave in their minds at this point is the relation between the reality and measurements.
The reality exists independent of any measurement and measurements are relative and uncertain, without that presumption, any science, including not only natural sciences but also social sciences, would become non-science.
Special-Student-7-Rebutter
While there are many people who claim that they do not need to study mathematics ("because mathematics is not any useful for their daily lives", they say), there are some things understood best (or probably only) via mathematics.
References
<The previous article in this series | The table of contents of this series |
<The previous article in this series | The table of contents of this series | The next article in this series>
, while real numbers are as imaginary as imaginary numbers
Topics
About:
high school mathematics
The table of contents of this article
Starting Context
Target Context
-
The reader will know how the complex numbers system models the reality.
Orientation
There is an article on becoming a benefactor of humanity by being a conduit of truths
Main Body
1: Are Imaginary Numbers Imaginary?
Special-Student-7-Hypothesizer
Any imaginary number is \(b i\) where \(b\) is any real number and \(i\) is the number such that \(i^2 = - 1\).
There are some (maybe most) people who believe that imaginary numbers are "imaginary".
Special-Student-7-Rebutter
That is a bad name ...
Special-Student-7-Hypothesizer
The objection is natural: "The name is suggesting that they are imaginary ...".
But the name is a remnant of a historical ignorance.
In fact, misnomers are not so rare even in mathematics; "congruent" is one of them.
Special-Student-7-Rebutter
What "ignorance"?
Special-Student-7-Hypothesizer
Probably, the concept of 'imaginary number' was 1st conceived in the attempt to solve cubic equations.
Special-Student-7-Rebutter
"cubic"? instead of quadratic?
Special-Student-7-Hypothesizer
Probably so: you may think that the attempt to solve \(x^2 = - 1\) caused \(i\), but \(x^2 = - 1\) was just "It has no solution" without any further development.
But Cardano's method for cubic equations required \(\omega := (- 1 + \sqrt{3} i) / 2\), which is a cubic root of \(1\): you will see that \(\omega\) is indeed a cubic root of \(1\) by diligently expanding \(\omega^3\) with \(i^2 = - 1\).
But \(i\) at that time was just an expedient: "There is not such a thing, but if we pretend that it existed, ta-da!, we can solve cubic equations.".
Special-Student-7-Rebutter
"ta-da"?
Special-Student-7-Hypothesizer
Anyway, it was thought imaginary, meaning that "it does not really exist.".
Special-Student-7-Rebutter
To be accurate, it does not exist in the real numbers set.
Special-Student-7-Hypothesizer
Certainly, but the people at that time could not imagine beyond the real numbers set, and so, as far as it does not exist in the real numbers set, "it did not exist at all" for them.
Special-Student-7-Rebutter
Just as "\(\sqrt{2}\) did not exist at all" for the people who could not imagine beyond the rational numbers set.
Special-Student-7-Hypothesizer
Nothing was wrong with \(i\), but something was wrong with the people's imagination.
Special-Student-7-Rebutter
In the 1st place, what do "real" and "imaginary" mean?
2: Does Mathematics Discover or Invent?
Special-Student-7-Hypothesizer
In order to think of that, we should think of the question, "Does mathematics discover or invent?".
Special-Student-7-Rebutter
Ah, that is a well-heard question.
Special-Student-7-Hypothesizer
While some people have expressed their own stances, I am pretty sure on that question: mathematics is a model of the reality.
Special-Student-7-Rebutter
So, ..., does it discover or invent?
Special-Student-7-Hypothesizer
Any model itself is a human-made artifact, but mathematics is not something arbitrarily trumped-up but something that aspires to appropriately represent the reality.
Special-Student-7-Rebutter
Cannot a branch of mathematics not aspire so?
Special-Student-7-Hypothesizer
It could, but any branch gets some significant attention only when it represents the reality well, because otherwise, the branch would have no application.
Special-Student-7-Rebutter
So, ..., does mathematics discover or invent?
Special-Student-7-Hypothesizer
Mathematics models the reality, I am saying.
In fact, I do not think so meaningful to categorize it in "discover" or "invent": it is in "discover" in order to model the reality and is also in "invent" because it is a human-artifact.
3: Then, What Do "Real" and "Imaginary" Mean?
Special-Student-7-Hypothesizer
If "real" means being an entity in the reality, any concept in mathematics is not real, because the concept is an entity in the human-made model, not in the reality.
So, 'real numbers' will be no more real than 'imaginary numbers'.
Special-Student-7-Rebutter
It is not that real numbers exist in the reality.
Special-Student-7-Hypothesizer
But the real numbers set models the reality well.
In fact, the real numbers set models any line well.
Special-Student-7-Rebutter
It is not only spacial lines that the real numbers set models.
Special-Student-7-Hypothesizer
Certainly, for example, also masses of objects are modeled by real numbers.
Anyway, the real numbers set is regarded to be real because it models the reality well.
So, "real" is appropriately understood as modeling the reality well.
So, whether 'imaginary number' is real is the matter of whether it models the reality well.
4: How Complex Numbers Set Models Reality
Special-Student-7-Hypothesizer
The reason why 'imaginary number' was regarded to be non-real is that the people thought that it did not represent anything in the reality.
In fact, what line segment has the i length?
Special-Student-7-Rebutter
But a concept does not need to model a line but to model an entity in the reality.
Special-Student-7-Hypothesizer
In fact, we should think of the complex numbers set, of which the imaginary numbers set is a part.
The complex numbers set models any 2-dimensional plane.
For any 2-dimensional plane, any point is represented by a complex number, \(x + y i\), and any complex number represents a point on the plane; technically, the points set on the plane corresponds to the complex numbers set bijectively.
Then, \(i\) represents the \(0 + 1 i\) point on the plane.
Regarding \(x + y i\) as the vector from the origin to \(x + y i\), \(x + y i = r (cos \theta + sin \theta) i\), where \(r\) is the length of the vector and \(\theta\) is the counter-clock-wise-angle of the vector from the x-axis, as is well known.
Let us define \(e^{\theta i} := cos \theta + sin \theta i\).
Special-Student-7-Rebutter
That definition is possible only because \(e^{0 i} := cos 0 + sin 0 i = 1 = e^0\).
Special-Student-7-Hypothesizer
Well, I could insist that the exponential function for real numbers and the exponential function for imaginary numbers were totally different creatures with the same symbol happened to be shared, but that would be a bad practice, so, yes, we use the same symbol only because the 2 exponential functions are seamlessly connected.
Anyway, that is just a definition, because the exponential map with imaginary number arguments had not been defined as far as we were concerned.
Special-Student-7-Rebutter
The essence of the complex numbers set is not only that the set bijectively corresponds to the plane points set, set-wise.
In fact, what does \(i^2 = - 1\) mean?
Special-Student-7-Hypothesizer
Thinking of \(i^2\) means thinking of multiplication of complex numbers.
As the real numbers set allows the arithmetic operations, the complex numbers set allows the arithmetic operations, \((x + y i) + (x' + y' i) = (x + x') + (y + y') i\); \((x + y i) - (x' + y' i) = (x - x') + (y - y') i\); \((x + y i) (x' + y' i) = (x x' - y y') + (x y' + y x') i\); \((x + y i) / (x' + y' i) = ((x + y i) (x' - y' i)) / (x'^2 + y'^2) = ((x x' + y y') + (- x y' + y x') i) / (x'^2 + y'^2) = (x x' + y y') / (x'^2 + y'^2) + (- x y' + y x') / (x'^2 + y'^2) i\), where \(x' + y' i \neq 0\).
Special-Student-7-Rebutter
An important fact is that that is an expansion of the arithmetic operations for the real numbers set.
Special-Student-7-Hypothesizer
Yes, we have defined the arithmetic operations for the complex numbers set as they are consistent with the arithmetic operations for the real numbers set with the real numbers set as the subset of the complex numbers set.
Special-Student-7-Rebutter
So, the real numbers set sits seamlessly in the complex numbers set.
Special-Student-7-Hypothesizer
And the amazing fact is that \(e^{(\theta + \theta') i} = e^{\theta i} e^{\theta' i}\) holds: \(e^{(\theta + \theta') i} := cos (\theta + \theta') + sin (\theta + \theta') i\) while \(e^{\theta i} e^{\theta' i} = (cos \theta + sin \theta i) (cos \theta' + sin \theta' i) = (cos \theta cos \theta' - sin \theta sin \theta') + (cos \theta sin \theta' + sin \theta cos \theta') i = cos (\theta + \theta') + sin (\theta + \theta') i\).
And the exponential function for complex numbers can be seamlessly defined as \(e^{x + y i} := e^x e^{y i}\), satisfying \(e^{(x + y i) + (x' + y' i)} = e^{x + y i} e^{x' + y' i}\).
Special-Student-7-Rebutter
The definition of the exponential function for complex numbers is just a definition, but the amazing fact is that the function has become seamless with the exponential function for real numbers and satisfies the expected property.
Special-Student-7-Hypothesizer
As each \(x + y i\) is regarded to be a vector, the addition of any 2 complex numbers is the addition of the corresponding 2 vectors, as is obvious.
The product of any 2 complex numbers is the vector whose length is the product of the lengths of the 2 vectors and whose angle is the sum of the angles of the 2 vectors, which is because \((x + y i) (x' + y' i) = r e^{\theta i} r' e^{\theta' i} = r r' e^{(\theta + \theta') i}\).
Special-Student-7-Rebutter
So, the complex numbers set models the plane in the reality well.
Special-Student-7-Hypothesizer
So, the complex numbers set is no less real than the real numbers set.
5: Thinking Imaginary Numbers Imaginary Is a Sure Sign of Narrow-Mindedness
Special-Student-7-Hypothesizer
So, \(i\) is the \((0, 1)\) point on the plane and \(i^2 = - 1\) is reasonably understood as the result of the multiplication: \(i^2 = i i\) has the length, \(1 \times 1 = 1\), and the angle, \(\pi / 2 + \pi / 2 = \pi\), which is nothing but \(- 1\).
So, \(i^2 = - 1\) is no mystery at all.
Special-Student-7-Rebutter
Someone who thinks that \(i\) is imaginary is a 1-dimensional creature that is caged in a line and cannot see or imagine the existence of the world outside the line.
Special-Student-7-Hypothesizer
The word, "imaginary number", was coined at the time when the people did not understand how imaginary numbers modeled the reality, but much time has passed, many things have developed, and you should not be saying like "Imaginary numbers are imaginary." any more.
I heard an argument that wave function in quantum mechanics was not real because it was 'complex numbers'-valued, but that is a sure sign of ignorance.
Special-Student-7-Rebutter
Of course, everyone (especially, we) is ignorant more or less, and it will be fine if you humbly admit your ignorance and continually rectify your misconceptions.
6: Why We Think of 'Field'
Special-Student-7-Hypothesizer
The complex numbers set with the arithmetic operations constitutes a field, as the real numbers set with the arithmetic operations constitutes a field.
Also the rational numbers set with the arithmetic operations is a field, but the integers set with the addition, the subtraction, and the multiplication is not any field but a ring, which is because it does not have any division, which means that \(1 / 2\) does not belong to the integers set.
Special-Student-7-Rebutter
As has been mentioned in What Is a Vector?, it is important to think of the space.
Special-Student-7-Hypothesizer
You may think like "The division operation exists in the integers set, because we can do '1 / 2'", but that is not on the integers set because the result is not in the integers set.
Special-Student-7-Rebutter
Anyway, why should we introduce such an abstract concept as 'field'?
Special-Student-7-Hypothesizer
A major reason is, that is for the sake of economy.
In fact, as far as a structure is a field, some many conclusions hold for the structure regardless of that it is the rational numbers field, the real numbers field, the complex numbers field, or any other field.
So, it is very uneconomical to state and prove the conclusions individually for each of the various fields.
There seem some people who show sheer rejections to abstract concepts, but do you really want to individually deal with each of the various fields?
If you begin to individually study various fields, probably (unless you are too unintelligent), you will begin to feel studying essentially same things again and again, and probably (unless you are too unintelligent), you will find it unwise.
On the other hand, once you accept the concept of 'field', you can just study 'field' and can make same conclusions for all the fields.
Which do you prefer?
7: Why Not Higher-Dimensional Numbers Systems?
Special-Student-7-Rebutter
The complex numbers system (field) is fine modeling the 2-dimensional plane, but then, a natural question will be "Are there a numbers system for the 3-dimensional space, a numbers system for the 4-dimensional space, etc.?".
Special-Student-7-Hypothesizer
That question is "While the complexes numbers system exists, why not a numbers system like \(x + y i + z j\)? or does it exist?".
The answer is that it does not exist as a field.
Special-Student-7-Rebutter
Why?
Special-Student-7-Hypothesizer
If you define it just as a vectors space, it is of course possible: let \(x + y i + z j\) represent the \((x, y, z)\) point.
But it is not possible to define any set of arithmetic operations on it to make it a field.
Special-Student-7-Rebutter
Why?
Special-Student-7-Hypothesizer
It is just impossible: supposing that it expands the complex numbers field, setting \(i j = a + b i + c j\) where \(a, b, c\) are some real numbers, \(i i j = i (a + b i + c j)\), but the left hand side is \(i^2 j = - j\) and the right hand side is \(a i + b i^2 + c i j = a i - b + c i j = a i - b + c (a + b i + c j) = a c - b + (a + b c) i + c^2 j\), which means that \(a c - b = 0\), \(a + b c = 0\), and \(c^2 = - 1\), which is impossible because \(c^2 = - 1\) is impossible.
Special-Student-7-Rebutter
What if it does not expand the complex numbers field?
Special-Student-7-Hypothesizer
Usually, it is useful only because it expands the complex numbers field, as the complex numbers field is useful only because the complex numbers field has expended the real numbers field.
Special-Student-7-Rebutter
Anyway, what if?
Special-Student-7-Hypothesizer
Anyway, it should be impossible, although I do not show the proof here.
Special-Student-7-Rebutter
Fascinating! Why are the 1-dimensional space and the 2-dimensional space so special to allow the field structures?
Special-Student-7-Hypothesizer
Probably, the concept of 'field' is so special to choose only the 1-dimensional space and the 2-dimensional space.
Special-Student-7-Rebutter
What if the structure is not required to be 'field'?
Special-Student-7-Hypothesizer
In fact, there is 'quaternion' for the 4-dimensional space, which is not any field but an associative division algebra over the real numbers field, where "division" means that each nonzero element has the multiplicative inverse.
Special-Student-7-Rebutter
How is that "associative division algebra" thing different from 'field'?
Special-Student-7-Hypothesizer
That thing is not commutative: \(i j = k\) but \(j i = - k\), for example.
Special-Student-7-Rebutter
But there is no 3-dimensional associative division algebra over the real numbers field?
Special-Student-7-Hypothesizer
Certainly.
Special-Student-7-Rebutter
Fascinating! The 3-dimensional space has been skipped over by the structure.
Special-Student-7-Hypothesizer
It is interesting that while an \(m\)-dimensional Euclidean space and an \(n\)-dimensional Euclidean space do not seem so different as far as vectors space structure is concerned, they can be so different as far as field structure or associative division algebra structure is concerned.
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
"A quantity that has both size and direction"? But what is 'size' or 'direction'? In fact, any vector is an element of a vectors space, which is not any tautology.
Topics
About:
high school mathematics
The table of contents of this article
Starting Context
Target Context
-
The reader will know what a vector is.
Orientation
There is an article on becoming a benefactor of humanity by being a conduit of truths
There is an article on definition of set and Russel's "paradox".
Main Body
1: A Vector Is "a Quantity That Has Both Size and Direction"?
Special-Student-7-Hypothesizer
The most prevalent rough definition of 'vector' seems to be "any quantity that has both size and direction".
Special-Student-7-Rebutter
That is not any valid definition unless "size" and "direction" are defined.
Special-Student-7-Hypothesizer
'size' is ... a size, you know.
Special-Student-7-Rebutter
I do not know, actually.
What is 'size' of a book? The length?, the width?, the thickness?, the weight?, the number of pages?, the number of words?, the amount of the information contained in it?, the price?, or what? Or why cannot I just declare it to be 42?
Special-Student-7-Hypothesizer
I do not know why you cannot: if you define the size to be 42, the size should be 42.
Special-Student-7-Rebutter
Then, I do not understand what "has size" means, because anything can be regarded to "have size" by just our regarding it to have "size 42".
Special-Student-7-Hypothesizer
That seems true.
Special-Student-7-Rebutter
And what is 'direction'?
Special-Student-7-Hypothesizer
In the 3-dimensional space, you know what 'direction' is.
Special-Student-7-Rebutter
For the 3-dimensional space case, yes, I have some intuition, but that is not any general definition.
Special-Student-7-Hypothesizer
Well, when the quantity is a combination of 4 real numbers like (1.3, 2.5, 0.9, 4.2), the direction is the proportion of the 4 numbers: (1.3, 2.5, 0.9, 4.2) and (1.3 * 2, 2.5 * 2, 0.9 * 2, 4.2 * 2) have the same direction.
Special-Student-7-Rebutter
Is any vector always a combination of some real numbers?
Special-Student-7-Hypothesizer
That is not the case.
Special-Student-7-Rebutter
Then, that is just an example, not any general definition.
Special-Student-7-Hypothesizer
That is true.
2: Enter 'Scalar Multiplication'!
Special-Student-7-Hypothesizer
In fact, we need to introduce the concept of 'scalar multiplication' in order to begin to talk about 'direction'.
We introduce a scalars set, which is a field, and introduce a scalar multiplication, which means multiplying a vector by an element of the scalars set to get a vector.
Any field is, very roughly speaking, a structure that allows a set of arithmetic operations, (+, -, *, /), and a typical example is the real numbers set, \(\mathbb{R}\).
Just for the sake of simplicity, in this article, we will assume that the scalars set is always the real numbers set unless specified otherwise.
Any vector, v, can be multiplied by any real number, r, and the result is r v.
In the above example, v = (1.3, 2.5, 0.9, 4.2), r = 2, and r v = (1.3 * 2, 2.5 * 2, 0.9 * 2, 4.2 * 2).
And for any fixed v, r v with any r has the same direction.
On the other hand, for some 2 vectors, v and v', if there is no r such that v' = r v, v and v' have some different directions.
You can say that any direction is an equivalence class of vectors with respect to the equivalence relation, v ~ v' if and only if v = r v'.
Special-Student-7-Rebutter
When r < 0, do v and r v have the same direction?
Special-Student-7-Hypothesizer
Well, that is a matter of the definition of the word, and we can say that they have the same direction, while colloquially, they are usually said to have "opposite directions" (the problem of calling it "opposite directions" is that the scalars set is not necessarily the real numbers set and the scalars set does not have necessarily have the concept of "plus" or "minus").
3: Vector Is an Element of a Vectors Space
Special-Student-7-Rebutter
Anyway, a moral is that it is meaningless to talk about whether a single object has a direction by itself: we need to think of a space of objects on which a scalar multiplication is defined, and only by dividing the space into equivalence classes, the concept of direction arises.
Without 1st defining the space, we cannot talk about whether an object "has a direction", so, it is meaningless to talk about whether a single object is a vector by itself.
Special-Student-7-Hypothesizer
So, we say that any vector is an element of a vectors space.
Someone may say that that is a tautology, but no, we 1st define a space and call the space a vectors space, and then, an element of the space is called "vector".
A rigorous definition of vectors space is here.
Special-Student-7-Rebutter
The way of thinking is important, because it is prevalent in mathematics.
For example, mathematics talks about spaces (or called "structures") like groups, rings, fields, modules, etc., and a group element is an element of a group: it is meaningless to talk about a group element by itself.
The prevalent pattern is, we take a set and take some operations on the set with some rules, and the set with the operations is a space (structure).
'vectors space' is one of such spaces and a vector is an element of such a space.
4: What 'Size' Really Means
Special-Student-7-Rebutter
That rigorous definition does not mention 'size' at all.
Special-Student-7-Hypothesizer
In fact, a vector does not need to have any absolute size.
Special-Student-7-Rebutter
What do you mean by "absolute size"?
Special-Student-7-Hypothesizer
For a nonzero vector, v, and a positive scalar, r, the "size" of r v could be said to be r-times the "size" of v.
So, we could talk about the relative sizes of some 2 vectors in the same direction, but comparing the sizes of some 2 vectors in some different directions is not meaningful in general.
For example, think of a vectors space, in which any element is the tuple of the length (in cm) and the weight (in kg) of an object.
Special-Student-7-Rebutter
Strictly speaking, that is not any vectors space, because each negative scalar multiple of any vector must exist in the space.
Special-Student-7-Hypothesizer
Then, think of differences of 2 lengths and differences of 2 weights, which can be negative.
Special-Student-7-Rebutter
All right.
Special-Student-7-Hypothesizer
Is the size of a vector, (180, 70), meaningful?
Special-Student-7-Rebutter
I, personally, find it not so meaningful.
Special-Student-7-Hypothesizer
And which is larger between (185, 65) and (175, 75)?
Special-Student-7-Rebutter
Someone may say that (175, 75) < (185, 65) claiming that length is more important, another one may say that (185, 65) < (175, 75) claiming that weight is more important, and yet another one may compare between \(185^2 + 65^2\) and \(175^2 + 75^2\), while I, personally, do not believe that \(185^2 + 65^2\) is particularly meaningful (for example, if we take length in mm instead, the comparison can be changed).
Special-Student-7-Hypothesizer
Or while the set of at-most-n-degree real polynomials canonically constitutes a vectors space, what is the "size" of a polynomial?
Special-Student-7-Rebutter
You could define a "size", for example, as the square root of the sum of the squared coefficients, or as the absolute value of the n-degree coefficient, or as uniformly 42, why not?
Special-Student-7-Hypothesizer
A vector does not have any intrinsic "size" in general, even if we could compare the sizes of some 2 vectors in the same direction.
Special-Student-7-Rebutter
What if the scalars set is not the real numbers set?
Special-Student-7-Hypothesizer
Then, even talking about the relative sizes of some 2 vectors in the same direction may be meaningless.
Special-Student-7-Rebutter
Then, what does "any quantity that has both size and direction" mean?
Special-Student-7-Hypothesizer
"has size" seems to mean that a vector can be multiplied by scalars to become different vectors in the same direction.
Special-Student-7-Rebutter
There is the concept of 'norm' and you need to be careful not to confuse 'norm' with "size" meant in the meaning mentioned above.
Special-Student-7-Hypothesizer
In fact, "size" usually meant is really 'norm', but a vectors space does not need to be equipped with a norm.
I guess that "size" in "any quantity that has both size and direction" is prevalently understood as 'norm', but in that meaning, "vector is any quantity that has both size and direction" is wrong.
5: How Is the Set of the n-Tuples of Real Numbers Related to a Vectors Space?
Special-Student-7-Hypothesizer
Someone may think that any n-dimensional real vectors space is nothing but the set of the n-tuples of real numbers.
That is not exactly true, although any n-dimensional real vectors space can be represented by the set of the n-tuples of real numbers.
Special-Student-7-Rebutter
What does "be represented" mean?
Special-Student-7-Hypothesizer
Any n-dimensional real vectors space has a basis with some n elements, \({b_1, ..., b_n}\), each vector, v, can be expressed as a linear combination of the basis elements, \(v = v^1 b_1 + ... + v^n b_n\), and the vector can be represented by the n-tuple of the coefficients, \((v^1, ..., v^n)\).
"be represented" means that there is the 1-to-1 (bijective) linear mapping from the vectors space onto the set of the n-tuples of real numbers (technically, it is called that the vectors space is 'vectors spaces - linear morphisms' isomorphic to the set of the n-tuples of real numbers regarded as a vectors space).
Special-Student-7-Rebutter
Why does someone say as though any n-dimensional real vectors space is nothing but the set of the n-tuples of real numbers?
Special-Student-7-Hypothesizer
2 vectors spaces' being 'vectors spaces - linear morphisms' isomorphic means that the 2 vectors spaces have the same structure as vectors spaces, and the 2 vectors spaces can be handled in the same way as far as the vectors space operations are concerned.
Special-Student-7-Rebutter
"can be handled in the same way" does not mean that the 2 entities are the same entity.
Special-Student-7-Hypothesizer
Exactly, but some people like wordings like "the 2 spaces are the same", because probably those wordings bring about some labor-saving of expressions because expressions do not need to distinguish the 2 things, but at the core, we need to be aware that the 2 spaces are different.
For example, when we think of the change of the components of a vector with respect to a change of bases, unless you understand the existence of the vectors space independent of any representation, the concept would not make sense: as there is the vectors space independent of any representation, you return from a representation to the vectors space and go from the vectors space to another representation.
Not being able to distinguish between 'represented' and 'representation' seems a serious problem somehow prevalently seen.
6: What Is the Dimension of a Vectors Space?
Special-Student-7-Rebutter
While we have talked about "n-dimensional real vectors space", what is 'dimension' of a vectors space?
Special-Student-7-Hypothesizer
'dimension' of any vectors space is nothing but the number of the elements of any basis for the vectors space.
Special-Student-7-Rebutter
In the 1st place, does every vectors space have a basis?
Special-Student-7-Hypothesizer
In fact, if you do not conceive that question, you are not really learning mathematics.
And the answer is yes, although we do not show any proof here.
Special-Student-7-Rebutter
And do all the possible bases of a vectors space have the same number of elements?
Special-Student-7-Hypothesizer
Also that question is one that if you do not conceive, you are not really learning mathematics.
And the answer is yes, although we do not show any proof here.
Let us know that a vectors space may be infinite-dimensional.
Special-Student-7-Rebutter
What does that mean?
Special-Student-7-Hypothesizer
That of course means that the number of the elements of any basis for the vectors space is infinite, but note that that does not mean that a vector is expressed as an infinite linear combination of basis elements: any vector is expressed as a finite linear combination of basis elements.
Special-Student-7-Rebutter
Then, why does the basis need some infinite number of elements?
Special-Student-7-Hypothesizer
A vector is a linear combination of a set of some finite number of basis elements, another vector is a linear combination of another set of some finite number of basis elements, and in order for every vector to be a linear combination of its own set of some finite number of basis elements, the basis needs some infinite number of elements.
Special-Student-7-Rebutter
While usually the space around us is said to be 3-dimensional, does that mean that the space is a 3-dimensional vectors space?
Special-Student-7-Hypothesizer
Aside from the fact that the spacetime is said to be 4-dimensional, strictly speaking, no, the space is not any 3-dimensional vectors space, which is because the space is said to be bounded.
Special-Student-7-Rebutter
Ah, so, the dimension aside, the space is not even any vectors space.
Special-Student-7-Hypothesizer
The space (or the spacetime) is modeled as a manifold, and the dimension of manifold has another definition, although it is somehow related with dimension of vectors space.
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
Japanese schools talk about "shugo"s, but are they sets or collections?
Topics
About:
elementary school mathematics
The table of contents of this article
Starting Context
Target Context
-
The reader will know the distinction between sets and collections.
Orientation
There is an article on becoming a benefactor of humanity by being a conduit of truths
There is an article on definition of set and Russel's "paradox".
Main Body
1: What Does "shugo" (Used in Japanese Schools) Mean?
Special-Student-7-Hypothesizer
Japanese schools talk about "shugo" s.
For example, they think of the "shugo" of some 3 apples, the "shugo" of the students of a class, etc..
Special-Student-7-Rebutter
What is "shugo" exactly?
Special-Student-7-Hypothesizer
Any "shugo" is any collection of some objects such that the membership is unambiguous.
Special-Student-7-Rebutter
Is there ever a collection such that the membership is ambiguous?
Special-Student-7-Hypothesizer
I do not think there is: the concept of 'collection' itself includes the requirement that the membership is unambiguous; the qualification is just for emphasis.
Special-Student-7-Rebutter
So, is "shugo" nothing but 'collection' in English?
Special-Student-7-Hypothesizer
I can just guess so, because Japanese textbooks do not say what "shugo" is called in English, but dictionaries usually say that "shugo" in mathematics is "set" in English.
Special-Student-7-Rebutter
But 'set' and 'collection' are 2 different things in mathematics.
2: Caution: There Are Some Multiple Set Theories
Special-Student-7-Hypothesizer
As a caution, there are some multiple set theories, and what 'set' is in each theory depends on the theory.
Special-Student-7-Rebutter
Very confusing; cannot the theories use different names?
Special-Student-7-Hypothesizer
It is very confusing, but as each theory exists because it regards the other theories unsatisfactory, each theory says "'set' we call is the real 'set'!", and each theory uses "set".
Special-Student-7-Rebutter
I kind of understand, but ...
Special-Student-7-Hypothesizer
Anyway, arguably the most popular one is the ZFC set theory, and we will mean the ZFC set theory by "the set theory" and mean 'set' in the ZFC set theory by "set" hereafter.
Note that what we are going to say here are not particularly canonical; as we have not seen any convincing argument, we are presenting a hypothesis that is convincing for us. Nobody should accept something just because someone (whoever he or she is) says so or just because many people say so; check yourself whether the hypothesis is unboundedly consistent, which is the only way to approach truths.
3: How Have 'Set' and 'Collection' Ended Up Meaning Different Things?
Special-Student-7-Hypothesizer
Most textbooks cite Russel's "paradox" as the reason why the set theory needs to be as it is, but the arguments are not convincing at all, at least for us.
Special-Student-7-Rebutter
What are the arguments like?
Special-Student-7-Hypothesizer
Historically, there was the so-called "naive set theory", which is based on the so-called "naive comprehension axiom", which is "Any precisely specified property of member can be used to define a set.".
Then, Russel's "paradox" came along and refuted the "naive comprehension axiom", and so refuted the "naive set theory".
So, now, the ZFC set theory admits only the empty set and the things constructed from the empty set as "set".
So, the collection of some 2 electrons are not admitted to be "set", because the collection is not constructed from the empty set.
Special-Student-7-Rebutter
There seems a wide gap in the reasoning: why would the collection of the 2 electrons not be "set" just because the "naive comprehension axiom" has been refuted?
Special-Student-7-Hypothesizer
I have not seen any convincing explanation.
In fact, why will we not have the modified axiom that "Any precisely specified property of member that (the property) determines membership unambiguously can be used to define a set."?
Special-Student-7-Rebutter
What Russel's "paradox" says is that just "precisely specified property of member" does not guarantee the unambiguous-ness of membership, so, why will we not add the unambiguous-ness as an additional requirement?
Special-Student-7-Hypothesizer
I have not seen any convincing argument why; it seems just that as checking it is not easy in general, they want a way to be able to define sets without bothering to do checking.
Special-Student-7-Rebutter
Well, is that an attitude to humbly approach truths? That seems an attitude to ignore whatever are inconvenient for humans.
Special-Student-7-Hypothesizer
Whether the unambiguous-ness can be checked easily is just a matter of human convenience, and the essence of the ZFC set theory is that it gave up covering the concept of "collection" and has decided to degenerate to deal with only convenient-for-humans kind of collections, which are now called "sets", which is my understanding.
In fact, I do not say at all that having a theory about such limited kind of collections is bad, but we need to be aware what the theory is really doing.
Special-Student-7-Rebutter
We need to avoid some confusions: "naive set theory" means the theory with the "naive comprehension axiom" not the concept of 'collection'; while the ZFC set theory excludes most collections like a collection of some 2 electrons, that is not because the collection of 2 electrons are inappropriate but because the set theory has degenerated.
Special-Student-7-Hypothesizer
The main cause of confusions is that the ZFC set theory keeps using the term, "set", while the concept of 'set' has degenerated from equaling 'collection' to 'convenient-for-humans collection'.
4: In Fact, a Collection in General Is Not Defined by Any Property of Member
Special-Student-7-Hypothesizer
In fact, I think that the "naive comprehension axiom" is more fundamentally problematic than Russel's "paradox".
The problem is that it tries to define any collection by a property of member.
For example, let us think of the collection of some balls I put into a bag.
I put a ball into the bag just because my groping hand happened to touch the ball, not because the ball is red or blue or something.
Special-Student-7-Rebutter
You mean that the collection is not defined by any property of ball, like being red or something?
Special-Student-7-Hypothesizer
Yes.
Special-Student-7-Rebutter
Your collection is usually defined by enumerating the balls, as the collection is finite: you cannot put infinitely many balls into the bag by groping.
Special-Student-7-Hypothesizer
Yes, when the collection is finite, there is the escape.
But what if the collection is infinite?
Special-Student-7-Rebutter
How will you define an infinite collection not by property of member?
Special-Student-7-Hypothesizer
If we ignore Relativity, let us take a \(\mathbb{R}^3\) Cartesian coordinates system for the Universe, and take the 1-radius open ball around each rational-coordinates point. Then, the numbers of the electrons in each open ball at a time, which is a subcollection of the natural numbers set, is my infinite collection.
Special-Student-7-Rebutter
You mean that the collection is not defined by any property of natural number, like being even or something?
Special-Student-7-Hypothesizer
Yes.
Special-Student-7-Rebutter
It does not seem to be really infinite, supposing that there are only some finite number of electrons in the Universe.
Special-Student-7-Hypothesizer
Then, let us take the pair of the center-coordinates and the number of the electrons, like ((1.2, 3.4, 5.6), 7).
Special-Student-7-Rebutter
Then, certainly, the collection is infinite.
What if we are not allowed to ignore Relativity?
Special-Student-7-Hypothesizer
Let us take a chart for the spacetime manifold, whose (the chart's) domain is homeomorphic to \(\mathbb{R}^4\), and let us do the same thing for that \(\mathbb{R}^4\).
Special-Student-7-Rebutter
So, you are saying that such collections exist in the reality but the "naive comprehension axiom" cannot grasp such collections.
Special-Student-7-Hypothesizer
In fact, that is also the problem of the "restricted comprehension axiom", which is in fact the same with the "naive comprehension axiom" except that the "restricted comprehension axiom" thinks of only the elements of an already-known-to-be-set collection.
Special-Student-7-Rebutter
Is "the number of the electrons in the open ball" not 'property of member'?
Special-Student-7-Hypothesizer
At least, the ZFC set theory does not admit such a property: the "restricted comprehension axiom" requires that the property is expressed as a formula that allows only the specified operators like \(\in\) and some already-known-to-be-set collections.
Special-Student-7-Rebutter
Did the "naive comprehension axiom" admit such a property?
Special-Student-7-Hypothesizer
Maybe, but such a usage of the term, "property of member", would be quite harmful (a too-far-fetched interpretation, I would say), in my opinion. Being "the number of the electrons in the open ball" is not any property of the natural number but is a property of the Universe: I mean, being "the number of the electrons in the open ball" is not about the natures of the natural number (like being even, prime, or something) but about the natures of the Universe.
Special-Student-7-Rebutter
Anyway, the "restricted comprehension axiom" does not allow your collection.
Special-Student-7-Hypothesizer
So, there is the collection in the reality, but the ZFC set theory refuses to cope with the collection.
5: "shugo" Means 'Collection'
Special-Student-7-Hypothesizer
So, "shugo" in Japanese textbooks means 'collection' not "set".
And when a Japanese student later hears of Russel's "paradox" and hears '"set" of some 2 electrons' refuted, the "shugo" of the 2 electrons is not refuted at all, because the collection of the 2 electrons is not refuted at all.
And when he or she later hears that "naive set theory" is inappropriate, the concept of "shugo" (the concept of 'collection') is not inappropriate at all, because the concept of 'collection' is not based on the naive comprehension axiom.
6: The Concept of 'Collection' Lives on
Special-Student-7-Hypothesizer
The concept of 'collection' is still valid and is indeed used in mathematics.
As a typical example, in the category theory, a category in general is not any set but a collection. For example, the category of all the sets, \(Set\), is a collection but not any set.
Special-Student-7-Rebutter
\(Set\) is prevalently called "class".
Special-Student-7-Hypothesizer
Yes, 'class' is a concept wider than 'set' and narrower than 'collection', but anyway, a class is not any set in general.
Special-Student-7-Rebutter
How is 'class' narrower than 'collection'?
Special-Student-7-Hypothesizer
Roughly speaking, any class is still constructed from the empty set and is still defined by property of member with a formula.
Anyway, if something was inappropriate just because it was not "set" in the ZFC set theory, the whole category theory would be inappropriate.
And "collection" is still frequently used in many mathematical textbooks: while a reader may wonder why "collection" is being used instead of "set", that is because the mentioned object is not or is not necessarily any "set" according to the ZFC set theory, and if 'collection' was inappropriate, such textbooks would be inappropriate, which is not the case.
7: Mathematics Is Not Exactly Built on the Set Theory
Special-Student-7-Hypothesizer
A rather prevalent misconception is that "whole mathematics is built on the set theory.".
Special-Student-7-Rebutter
That is what the naive set theory aspired to do.
Special-Student-7-Hypothesizer
But the attempt failed (by mainly Russel's "paradox") and because the set theory gave up dealing with general collections and has degenerated to be a theory on a very limited kind of collections, the aspiration is given up.
Certainly, the set theory is still a very important part of mathematics, but it is not exactly the basis of whole mathematics.
Special-Student-7-Rebutter
A proof is that "collection" is still used in many mathematical textbooks.
References
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